Irma Vazquez

The Intersection of Lesson Study and Design Research: A 3-D Visualization Development Project for the Elementary Mathematics Curriculum

This chapter describes how the research methods for developing a 3-D visualization program for elementary children closely resemble the principles of lesson study. While many lesson study experiences offer teachers opportunities for personal professional development to deepen their pedagogical content knowledge, this team’s immediate focus is on students’ mathematical understanding as they engage in research lesson activities that are the basis for development of new materials for the elementary mathematics classroom.

Connections to Numeracy

Volume of rectangular prisms extended into an interesting scaling problem: One child asked if they could try to build a Soma cube as large as the teacher’s demonstration Soma cube (made with very large individual cubes) using the smaller Soma figures. Initially they built four Soma cubes using the small sets of Soma figures and arranged them to look like a very large Soma figure #2. They eagerly attacked the problem of how many little individual cubes were in this figure each showing at least two ways to find the answer. The following week they built a huge cube using 27 smaller sets of Soma cubes and calculated 27 × 27 to find how many little individual cubes were in this model. Permutations within cake patterns: At the end of each year, the children were challenged to create a “cake” with the seven Soma figures. The cake had to have 24 cubes for a base and three “candles” on the second level. They created several 3 × 8 and 4 × 6 cakes and drew the top plan view coding patterns for these to submit to the baker. We share how children from three different cohorts discovered the permutations of each cake design from the three or four Soma figures (#1, #5, #6, and #7) that have the same 3-cube footprint.

Theoretical Frameworks and School Context

The rationale and Spatial Operational Capacity theoretical framework that support the project are described along with the design research methodology that evolved over the 7 years of research, the school context, and a pre-interview used to support the researchers’ perspectives on each participant child’s beginning spatial ability.

Introduction and Project Background

The project began as part of a PhD dissertation study in South Africa, by Retha Van Niekerk after she met with Pierre Van Hiele and with researchers within the Wiskobas project at the Freudenthal Institute in The Netherlands. Her work with Ton Lecluse, author of the Geocadabra Construction Box, developed for this project, and her professional association with authors, Sack and Vazquez, formed the basis for the work done in the US after she returned to South Africa. A brief overview of the book’s organization will end this chapter.

The Geocadabra Construction Box Dynamic Geometry Interface

The children learned to use the Geocadabra Construction Box by constructing virtual cube structures to match the conventional 2D figures in the printed Build and Explore with Geocadabra manual. The process builds proficiency with top-plan numeric views. As they progressed, they were able to digitally reproduce cube structures like those they constructed with loose cubes on the desktop next to their computers. Activities integrating top-plan numeric views and top, front and side views are shared.

3D to 2D via Top-View Plans

Having become proficient at correlating 2D pictures with top plan numeric view representations learners used the Construction Box to create their own task card puzzles for peers to solve without the aid of the digital interface. These included 2D pictures of Soma assemblies and later, top plan view diagrams that they drew by hand. These tasks required adept mental transformation skills moving among the SOC representations. The Extended Construction Box module was created to allow users to construct digital cube figures with holes or overhangs within the first octant of a 3D coordinate space. Learners then developed a top plan view coding system that allowed for holes or overhangs. They also used their knowledge of top plan views to represent rectangular prisms made up of unit cubes.